# Intuition behind conjugation in group theory

I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian groups), it seems pretty innocent, I have to admit that I have no intuition about whats happening, or why even such a map is important. Did anyone feel this way when you learnt it? Any examples/references that you have found useful when you learnt it?

Let’s make this precise. Let $G$ be a group and let $X$ be a set on which $G$ acts faithfully. For example, if $G$ is the Euclidean group of isometries of the plane, $X$ could be… well, the plane. Let’s say we have a reasonably concrete description of what a particular element $g \in G$ does to $X$. For example, if $G, X$ are as above, then perhaps $g$ is a rotation counterclockwise by $\theta$ centered at the origin.
Given that description, what does the element $hgh^{-1}$ do? Well, this is actually quite simple: in the description of $g$, replace all the elements $x \in X$ that occur by the corresponding elements $hx \in X$. For example, if $h$ is a translation by a vector $v$, then $hgh^{-1}$ is a rotation counterclockwise by $\theta$ centered at $v$ instead of the origin.
In other words, what conjugation does in terms of group actions (and it is always a good idea to think of groups in terms of their actions) is it corresponds to a “change of coordinates” on the underlying set $X$. This is a basic reason conjugation and group theory are important in understanding Newtonian mechanics (where $G$ is the Galilean group) or relativity (where $G$ is the Lorentz group) or really much of modern physics in general: in physics groups are always studied because of their actions and we always want our concepts and equations to be invariant under these actions. (For example, mass and charge are invariant concepts. The mass or charge of an object doesn’t change when you rotate or translate it.)
This gives a very intuitive definition of a normal subgroup: it’s a subgroup that “looks the same from every perspective.” For example, the subgroup of translations in the Euclidean group is always normal because the description “$g$ is a translation” is the same from every perspective (that is, it’s invariant under conjugation). It’s a good exercise to look at some groups you’re familiar with and see if you can identify which subgroups are normal based on this principle. (This principle underlies the importance of normal subgroups in Galois theory as well.)